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A room in the shape of a triangle has sides of length 7 yd, 8 yd, and 11 yd. If carpeting costs $17.50 a square yard and padding costs $3.25 a square yard, how much to the nearest dollar will it cost to carpet the room, assuming that there is no waste?
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MissPhiladelphia
First, you have to calculate the area of the triangle. The formula would be

[tex] A_{T} = \sqrt{s(s-a)(s-b)(s-c)} [/tex]

where s is the semi-parameter of the triangle with sides a, b and c

[tex]s= \frac{a+b+c}{2} [/tex]

[tex]s= \frac{7+8+11}{2}=13[/tex]

So,

[tex]A_{T} = \sqrt{13(13-7)(13-8)(13-11)}[/tex]

[tex]A_{T} = 27.93 \ yd^{2} [/tex]

So, the total cost of carpeting the room is,

[tex]27.93 \ yd^{2} (17.50 \ per \ yd^{2}+3.25 \ \ per \ yd^{2})=579.52[/tex]

$579.5
By sketching this triangle it can be determined that this is not a right triangle, but rather an oblique-angled triangle. For the lengths of the three sides, let a = 7, b = 8, and c = 11, and let angle A be opposite side a, and angle B be opposite side b, and angle C be opposite side c. 


Using the Law of Cosines: 

cos(A) = (b² + c² - a²)/2bc = [(8)² + (11)² - (7)²)/2(8)(11) = 0.773 

A = acos(0.773) = 39.4° 


Using the Law of Sines: 

sin(A)/a = sin(B)/b ==> sin(B) = (b/a)sin(A) = (8/7)sin(39.4°) = 0.725 

B = asin(0.725) = 46.5° 


Knowing angles A and B, angle C is: 

C = 180° - A - B = 180° - 39.4° - 46.5° = 94.1° 


Finally, the area is: 

Area = (ab/2)sin(C) = [(7)(8)/2]sin(94.1°) = 27.928 yd² 

The total cost is (27.928)($17.50 + $3.25) = $579.51

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